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In mathematics, a generalized inverse of a matrix A is a matrix that has some properties of the inverse matrix of A but not necessarily all of them. Formally, given a matrix and a matrix , is a generalized inverse of if it satisfies the condition .
The purpose of constructing a generalized inverse is to obtain a matrix that can serve as the inverse in some sense for a wider class of matrices than invertible matrices. A generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then this inverse is its unique generalized inverse. Some generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup.
Motivation for the generalized inverse
Consider the linear system
Suppose the matrix is singular, or . Then we need a right candidate of order such that for all ,
That is, is a solution of the linear system . Equivalently, we need a matrix of order such that
Hence we can define the generalized inverse as follows: Given an matrix , an matrix is said to be a generalized inverse of if
Construction of generalized inverse
The following characterizations are easy to verify:
- If is a rank factorization, then is a g-inverse of , where is a right inverse of and is left inverse of .
- If for any non-singular matrices and , then is a generalized inverse of for arbitrary and .
- Let be of rank . Without loss of generality, let
- where is the non-singular submatrix of . Then,
- is a g-inverse of .
Types of generalized inverses
The Penrose conditions are used to define different generalized inverses for and :
If satisfies the first condition, then it is a generalized inverse of . If it satisfies the first two conditions, then it is a generalized reflexive inverse of . If it satisfies all four conditions, then it is a Moore–Penrose pseudoinverse of .
Other kinds of generalized inverse include:
- One-sided inverse (left inverse or right inverse): If the matrix has dimensions and is full rank, then use the left inverse if and the right inverse if .
- Left inverse is given by , i.e., , where is the identity matrix.
- Right inverse is given by , i.e., , where is the identity matrix.
- Drazin inverse
- Bott–Duffin inverse
- Moore–Penrose pseudoinverse
with vector of unknowns and vector of constants, all solutions are given by
parametric on the arbitrary vector , where is any generalized inverse of . Solutions exist if and only if is a solution, that is, if and only if .
- Ben-Israel, Adi; Greville, Thomas N.E. (2003). Generalized inverses. Theory and applications (2nd ed.). New York, NY: Springer. ISBN 0-387-00293-6.
- Campbell, S. L.; Meyer, C. D. (1991). Generalized Inverses of Linear Transformations. Dover. ISBN 978-0-486-66693-8.
- Nakamura, Yoshihiko (1991). * Advanced Robotics: Redundancy and Optimization. Addison-Wesley. ISBN 0201151987.
- Rao, C. R.; Rao, C. Radhakrishna; Mitra, Sujit Kumar (1971). Generalized Inverse of Matrices and its Applications. New York: John Wiley & Sons. p. 240. ISBN 0-471-70821-6.
- Zheng, B; Bapat, R. B. (2004). "Generalized inverse A(2)T,S and a rank equation". Applied Mathematics and Computation. 155: 407–415. doi:10.1016/S0096-3003(03)00786-0.
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